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4037/13
ADDITIONAL MATHEMATICS
Paper 1
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Ordinary Level
October/November 2010
2 hours
Additional Materials:
*1581608333*
Answer Booklet/Paper
Graph Paper (1 sheet)
READ THESE INSTRUCTIONS FIRST
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Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
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Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Write your answers on the separate Answer Booklet/Paper provided.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of
angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 80.
This document consists of 5 printed pages and 3 blank pages.
DC (LEO/DJ) 25091/3
© UCLES 2010
[Turn over
2
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=
−b
b 2 − 4 ac .
2a
Binomial Theorem
(a + b)n = an +
()
()
where n is a positive integer and
()
n
n!
.
=
r
(n – r)!r!
2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
∆=
© UCLES 2010
()
n n–1
n n–2 2
n n–r r
a b+
a b +…+
a b + … + bn,
1
2
r
1
bc sin A
2
4037/13/O/N/10
3
1
Show that sec x – cos x = sin x tan x.
[3]
2
A 4-digit number is formed by using four of the seven digits 2, 3, 4, 5, 6, 7 and 8. No digit can be used
more than once in any one number. Find how many different 4-digit numbers can be formed if
(i) there are no restrictions,
[2]
(ii) the number is even.
[2]
3
The line y = mx + 2 is a tangent to the curve y = x2 + 12x + 18. Find the possible values of m.
[4]
4
The remainder when the expression x3 + kx2 – 5x – 3 is divided by x – 2 is 5 times the remainder when
the expression is divided by x + 1. Find the value of k.
[4]
5
Solve the simultaneous equations
log3 a = 2 log3 b,
log3 (2a – b) = 1.
[5]
6
Solve the equation 3x3 + 7x2 – 22x – 8 = 0.
7
(i) Sketch the graph of y = |3x – 5|, for –2  x  3, showing the coordinates of the points where the
graph meets the axes.
[3]
8
[6]
(ii) On the same diagram, sketch the graph of y = 8x.
[1]
(iii) Solve the equation 8x = |3x – 5|.
[3]
(a) A function f is defined, for x , by
f(x) = x2 + 4x – 6.
(i) Find the least value of f(x) and the value of x for which it occurs.
[2]
(ii) Hence write down a suitable domain for f(x) in order that f–1(x) exists.
[1]
(b) Functions g and h are defined, for x , by
g(x) =
x – 1,
2
h(x) = x2 – x.
(i) Find g–1(x).
[2]
(ii) Solve gh(x) = g–1(x).
[3]
© UCLES 2010
4037/13/O/N/10
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4
( )
1
9
(a) Find
2
x 3 – 3 dx.
[3]
(b) (i) Given that y = x
(ii) Hence find

x 2 + 6 , find
dy
.
dx
x2 + 3
dx.
x2 + 6
[3]
[2]
10 A particle travels in a straight line so that, t s after passing through a fixed point O, its displacement s m
from O is given by s = ln(t2 + 1).
11
(i) Find the value of t when s = 5.
[2]
(ii) Find the distance travelled by the particle during the third second.
[2]
(iii) Show that, when t = 2, the velocity of the particle is 0.8 ms–1.
[2]
(iv) Find the acceleration of the particle when t = 2.
[3]
Solve the equation
(i) 3 sin x – 4 cos x = 0, for 0°  x  360°,
[3]
(ii) 11 sin y + 1 = 4 cos2 y, for 0°  y  360°,
[4]
(
(iii) sec 2z +
© UCLES 2010
)
π
= –2, for 0  z  π radians.
3
4037/13/O/N/10
[4]
5
12 Answer only one of the following two alternatives.
EITHER
( )
π
A curve has the equation y = A sin 2x + B cos 3x. The curve passes through the point with coordinates
,3
12
π
and has a gradient of – 4 when x = .
3
(i) Show that A = 4 and find the value of B.
[6]
π
(ii) Given that, for 0  x  3 , the curve lies above the x-axis, find the area of the region enclosed by the
π
3
curve, the y-axis and the line x = .
[5]
OR
y
C
A
B
x
O
y = 4x2 – 2x3
The diagram shows the curve y = 4x2 – 2x3. The point A lies on the curve and the x-coordinate of A is 1.
The curve crosses the x-axis at the point B. The normal to the curve at the point A crosses the y-axis at the
point C.
(i) Show that the coordinates of C are (0, 2.5).
[5]
(ii) Find the area of the shaded region.
[6]
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reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of
Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2010
4037/13/O/N/10